LMFDB - Embedded newform 8100.2.d.q.649.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8100,2,Mod(649,8100)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8100.649"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8100.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,-6,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,-18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$64.6788256372$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.4057180416.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 9x^{6} + 22x^{4} + 12x^{2} + 1 $ x^8 + 9x^6 + 22x^4 + 12*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	no (minimal twist has level 900)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.6
Root			$-0.785261i$ of defining polynomial
Character	$\chi$	$=$	8100.649
Dual form			8100.2.d.q.649.3

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.680426i q^{7} -1.68043 q^{11} -5.14503i q^{13} -1.31957i q^{17} +0.324642 q^{19} -3.78924i q^{23} +8.64584 q^{29} -4.14503 q^{31} +1.35578i q^{37} -7.15009 q^{41} +7.29005i q^{43} -12.9654i q^{47} +6.53702 q^{49} +8.83052i q^{53} -8.81532 q^{59} +9.97048 q^{61} -4.17617i q^{67} -0.891185 q^{71} +7.82038i q^{73} -1.14341i q^{77} -9.65597 q^{79} -4.85659i q^{83} -17.4764 q^{89} +3.50081 q^{91} +8.93427i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{11} + 8 q^{19} - 18 q^{29} + 4 q^{31} - 18 q^{41} - 18 q^{49} - 30 q^{59} - 2 q^{61} - 24 q^{71} + 14 q^{79} - 6 q^{89} - 22 q^{91}+O(q^{100}) $ 8 * q - 6 * q^11 + 8 * q^19 - 18 * q^29 + 4 * q^31 - 18 * q^41 - 18 * q^49 - 30 * q^59 - 2 * q^61 - 24 * q^71 + 14 * q^79 - 6 * q^89 - 22 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8100\mathbb{Z}\right)^\times$.

$n$	$4051$	$6401$	$7777$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.680426i	0.257177i	0.991698	+	0.128588i	$0.0410446\pi$
$7$	0.680426i	0.257177i	−0.991698	+	0.128588i	$0.958955\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.68043	−0.506668	−0.253334	−	0.967379i	$-0.581527\pi$
$11$	−1.68043	−0.506668	−0.253334	+	0.967379i	$0.581527\pi$
$12$	0	0
$13$	− 5.14503i	− 1.42697i	−0.700669	−	0.713487i	$-0.747115\pi$
$13$	− 5.14503i	− 1.42697i	0.700669	−	0.713487i	$-0.252885\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.31957i	− 0.320044i	−0.987113	−	0.160022i	$-0.948844\pi$
$17$	− 1.31957i	− 0.320044i	0.987113	−	0.160022i	$-0.0511565\pi$
$18$	0	0
$19$	0.324642	0.0744779	0.0372390	−	0.999306i	$-0.488144\pi$
$19$	0.324642	0.0744779	0.0372390	+	0.999306i	$0.488144\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 3.78924i	− 0.790111i	−0.918657	−	0.395056i	$-0.870725\pi$
$23$	− 3.78924i	− 0.790111i	0.918657	−	0.395056i	$-0.129275\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.64584	1.60549	0.802746	−	0.596322i	$-0.203372\pi$
$29$	8.64584	1.60549	0.802746	+	0.596322i	$0.203372\pi$
$30$	0	0
$31$	−4.14503	−0.744469	−0.372234	−	0.928139i	$-0.621408\pi$
$31$	−4.14503	−0.744469	−0.372234	+	0.928139i	$0.621408\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.35578i	0.222890i	0.993771	+	0.111445i	$0.0355478\pi$
$37$	1.35578i	0.222890i	−0.993771	+	0.111445i	$0.964452\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.15009	−1.11666	−0.558328	−	0.829620i	$-0.688557\pi$
$41$	−7.15009	−1.11666	−0.558328	+	0.829620i	$0.688557\pi$
$42$	0	0
$43$	7.29005i	1.11172i	0.831275	+	0.555861i	$0.187611\pi$
$43$	7.29005i	1.11172i	−0.831275	+	0.555861i	$0.812389\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 12.9654i	− 1.89120i	−0.325333	−	0.945600i	$-0.605476\pi$
$47$	− 12.9654i	− 1.89120i	0.325333	−	0.945600i	$-0.394524\pi$
$48$	0	0
$49$	6.53702	0.933860
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.83052i	1.21297i	0.795097	+	0.606483i	$0.207420\pi$
$53$	8.83052i	1.21297i	−0.795097	+	0.606483i	$0.792580\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.81532	−1.14766	−0.573828	−	0.818976i	$-0.694542\pi$
$59$	−8.81532	−1.14766	−0.573828	+	0.818976i	$0.694542\pi$
$60$	0	0
$61$	9.97048	1.27659	0.638294	−	0.769792i	$-0.279640\pi$
$61$	9.97048	1.27659	0.638294	+	0.769792i	$0.279640\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 4.17617i	− 0.510200i	−0.966915	−	0.255100i	$-0.917892\pi$
$67$	− 4.17617i	− 0.510200i	0.966915	−	0.255100i	$-0.0821084\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.891185	−0.105764	−0.0528821	−	0.998601i	$-0.516841\pi$
$71$	−0.891185	−0.105764	−0.0528821	+	0.998601i	$0.516841\pi$
$72$	0	0
$73$	7.82038i	0.915307i	0.889131	+	0.457653i	$0.151310\pi$
$73$	7.82038i	0.915307i	−0.889131	+	0.457653i	$0.848690\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.14341i	− 0.130303i
$78$	0	0
$79$	−9.65597	−1.08638	−0.543191	−	0.839609i	$-0.682784\pi$
$79$	−9.65597	−1.08638	−0.543191	+	0.839609i	$0.682784\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 4.85659i	− 0.533080i	−0.963824	−	0.266540i	$-0.914119\pi$
$83$	− 4.85659i	− 0.533080i	0.963824	−	0.266540i	$-0.0858805\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−17.4764	−1.85249	−0.926245	−	0.376922i	$-0.876982\pi$
$89$	−17.4764	−1.85249	−0.926245	+	0.376922i	$0.876982\pi$
$90$	0	0
$91$	3.50081	0.366985
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.93427i	0.907137i	0.891221	+	0.453569i	$0.149849\pi$
$97$	8.93427i	0.907137i	−0.891221	+	0.453569i	$0.850151\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.8566	−1.08027	−0.540136	−	0.841578i	$-0.681627\pi$
$101$	−10.8566	−1.08027	−0.540136	+	0.841578i	$0.681627\pi$
$102$	0	0
$103$	− 11.6820i	− 1.15107i	−0.817778	−	0.575533i	$-0.804795\pi$
$103$	− 11.6820i	− 1.15107i	0.817778	−	0.575533i	$-0.195205\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.11550i	0.494534i	0.968947	+	0.247267i	$0.0795325\pi$
$107$	5.11550i	0.494534i	−0.968947	+	0.247267i	$0.920467\pi$
$108$	0	0
$109$	−7.17110	−0.686867	−0.343433	−	0.939177i	$-0.611590\pi$
$109$	−7.17110	−0.686867	−0.343433	+	0.939177i	$0.611590\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	17.6458i	1.65998i	0.557778	+	0.829990i	$0.311654\pi$
$113$	17.6458i	1.65998i	−0.557778	+	0.829990i	$0.688346\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.897873	0.0823078
$120$	0	0
$121$	−8.17617	−0.743288
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	3.13489i	0.278176i	0.990280	+	0.139088i	$0.0444172\pi$
$127$	3.13489i	0.278176i	−0.990280	+	0.139088i	$0.955583\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.11550	0.709055	0.354527	−	0.935046i	$-0.384642\pi$
$131$	8.11550	0.709055	0.354527	+	0.935046i	$0.384642\pi$
$132$	0	0
$133$	0.220895i	0.0191540i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 6.89119i	− 0.588754i	−0.955689	−	0.294377i	$-0.904888\pi$
$137$	− 6.89119i	− 0.588754i	0.955689	−	0.294377i	$-0.0951121\pi$
$138$	0	0
$139$	20.2555	1.71805	0.859023	−	0.511937i	$-0.171072\pi$
$139$	20.2555	1.71805	0.859023	+	0.511937i	$0.171072\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	8.64584i	0.723001i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.0674	−0.824750	−0.412375	−	0.911014i	$-0.635301\pi$
$149$	−10.0674	−0.824750	−0.412375	+	0.911014i	$0.635301\pi$
$150$	0	0
$151$	−18.7909	−1.52918	−0.764589	−	0.644518i	$-0.777058\pi$
$151$	−18.7909	−1.52918	−0.764589	+	0.644518i	$0.777058\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 19.4814i	− 1.55479i	−0.629015	−	0.777393i	$-0.716541\pi$
$157$	− 19.4814i	− 1.55479i	0.629015	−	0.777393i	$-0.283459\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.57830	0.203198
$162$	0	0
$163$	− 12.2640i	− 0.960589i	−0.877107	−	0.480294i	$-0.840530\pi$
$163$	− 12.2640i	− 0.960589i	0.877107	−	0.480294i	$-0.159470\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 13.5109i	− 1.04551i	−0.852483	−	0.522754i	$-0.824905\pi$
$167$	− 13.5109i	− 1.04551i	0.852483	−	0.522754i	$-0.175095\pi$
$168$	0	0
$169$	−13.4713	−1.03625
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 1.75465i	− 0.133404i	−0.997773	−	0.0667018i	$-0.978752\pi$
$173$	− 1.75465i	− 0.133404i	0.997773	−	0.0667018i	$-0.0212476\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−22.9393	−1.71457	−0.857283	−	0.514845i	$-0.827849\pi$
$179$	−22.9393	−1.71457	−0.857283	+	0.514845i	$0.827849\pi$
$180$	0	0
$181$	−12.4452	−0.925045	−0.462523	−	0.886607i	$-0.653056\pi$
$181$	−12.4452	−0.925045	−0.462523	+	0.886607i	$0.653056\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.21745i	0.162156i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.5785	−1.19958	−0.599788	−	0.800159i	$-0.704749\pi$
$191$	−16.5785	−1.19958	−0.599788	+	0.800159i	$0.704749\pi$
$192$	0	0
$193$	15.6147i	1.12397i	0.827147	+	0.561985i	$0.189962\pi$
$193$	15.6147i	1.12397i	−0.827147	+	0.561985i	$0.810038\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 17.8153i	− 1.26929i	−0.772804	−	0.634644i	$-0.781147\pi$
$197$	− 17.8153i	− 1.26929i	0.772804	−	0.634644i	$-0.218853\pi$
$198$	0	0
$199$	−8.07749	−0.572598	−0.286299	−	0.958140i	$-0.592425\pi$
$199$	−8.07749	−0.572598	−0.286299	+	0.958140i	$0.592425\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.88285i	0.412895i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.545537	−0.0377356
$210$	0	0
$211$	14.3887	0.990561	0.495281	−	0.868733i	$-0.335065\pi$
$211$	14.3887	0.990561	0.495281	+	0.868733i	$0.335065\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 2.82038i	− 0.191460i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−6.78924	−0.456694
$222$	0	0
$223$	2.03621i	0.136355i	0.997673	+	0.0681774i	$0.0217184\pi$
$223$	2.03621i	0.136355i	−0.997673	+	0.0681774i	$0.978282\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 27.3675i	− 1.81645i	−0.418485	−	0.908224i	$-0.637439\pi$
$227$	− 27.3675i	− 1.81645i	0.418485	−	0.908224i	$-0.362561\pi$
$228$	0	0
$229$	9.43001	0.623152	0.311576	−	0.950221i	$-0.399143\pi$
$229$	9.43001	0.623152	0.311576	+	0.950221i	$0.399143\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 0.191372i	− 0.0125372i	−0.999980	−	0.00626859i	$-0.998005\pi$
$233$	− 0.191372i	− 0.0125372i	0.999980	−	0.00626859i	$-0.00199537\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2.21076	−0.143002	−0.0715011	−	0.997441i	$-0.522779\pi$
$239$	−2.21076	−0.143002	−0.0715011	+	0.997441i	$0.522779\pi$
$240$	0	0
$241$	22.5075	1.44984	0.724918	−	0.688836i	$-0.241878\pi$
$241$	22.5075	1.44984	0.724918	+	0.688836i	$0.241878\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.67029i	− 0.106278i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−24.9654	−1.57580	−0.787901	−	0.615802i	$-0.788832\pi$
$251$	−24.9654	−1.57580	−0.787901	+	0.615802i	$0.788832\pi$
$252$	0	0
$253$	6.36754i	0.400324i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 16.3196i	− 1.01799i	−0.860770	−	0.508994i	$-0.830018\pi$
$257$	− 16.3196i	− 1.01799i	0.860770	−	0.508994i	$-0.169982\pi$
$258$	0	0
$259$	−0.922511	−0.0573221
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 16.6871i	− 1.02897i	−0.857499	−	0.514486i	$-0.827983\pi$
$263$	− 16.6871i	− 1.02897i	0.857499	−	0.514486i	$-0.172017\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3.36085	−0.204915	−0.102457	−	0.994737i	$-0.532671\pi$
$269$	−3.36085	−0.204915	−0.102457	+	0.994737i	$0.532671\pi$
$270$	0	0
$271$	20.0498	1.21794	0.608968	−	0.793195i	$-0.291584\pi$
$271$	20.0498	1.21794	0.608968	+	0.793195i	$0.291584\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 8.21238i	− 0.493434i	−0.969088	−	0.246717i	$-0.920648\pi$
$277$	− 8.21238i	− 0.493434i	0.969088	−	0.246717i	$-0.0793518\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0.376056	0.0224336	0.0112168	−	0.999937i	$-0.496430\pi$
$281$	0.376056	0.0224336	0.0112168	+	0.999937i	$0.496430\pi$
$282$	0	0
$283$	− 0.704881i	− 0.0419008i	−0.999781	−	0.0209504i	$-0.993331\pi$
$283$	− 0.704881i	− 0.0419008i	0.999781	−	0.0209504i	$-0.00666921\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 4.86511i	− 0.287178i
$288$	0	0
$289$	15.2587	0.897572
$290$	0	0
$291$	0	0
$292$	0	0
$293$	18.1914i	1.06275i	0.847136	+	0.531376i	$0.178325\pi$
$293$	18.1914i	1.06275i	−0.847136	+	0.531376i	$0.821675\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−19.4957	−1.12747
$300$	0	0
$301$	−4.96034	−0.285909
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4.04128i	0.230648i	0.993328	+	0.115324i	$0.0367906\pi$
$307$	4.04128i	0.230648i	−0.993328	+	0.115324i	$0.963209\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−9.16948	−0.519954	−0.259977	−	0.965615i	$-0.583715\pi$
$311$	−9.16948	−0.519954	−0.259977	+	0.965615i	$0.583715\pi$
$312$	0	0
$313$	− 17.2225i	− 0.973474i	−0.873549	−	0.486737i	$-0.838187\pi$
$313$	− 17.2225i	− 0.973474i	0.873549	−	0.486737i	$-0.161813\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 8.46298i	− 0.475328i	−0.971347	−	0.237664i	$-0.923618\pi$
$317$	− 8.46298i	− 0.475328i	0.971347	−	0.237664i	$-0.0763818\pi$
$318$	0	0
$319$	−14.5287	−0.813450
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 0.428389i	− 0.0238362i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	8.82200	0.486373
$330$	0	0
$331$	−7.03783	−0.386834	−0.193417	−	0.981117i	$-0.561957\pi$
$331$	−7.03783	−0.386834	−0.193417	+	0.981117i	$0.561957\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	24.5488i	1.33726i	0.743597	+	0.668629i	$0.233118\pi$
$337$	24.5488i	1.33726i	−0.743597	+	0.668629i	$0.766882\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6.96541	0.377198
$342$	0	0
$343$	9.21094i	0.497344i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 25.3263i	− 1.35958i	−0.733405	−	0.679792i	$-0.762070\pi$
$347$	− 25.3263i	− 1.35958i	0.733405	−	0.679792i	$-0.237930\pi$
$348$	0	0
$349$	10.8879	0.582817	0.291409	−	0.956599i	$-0.405876\pi$
$349$	10.8879	0.582817	0.291409	+	0.956599i	$0.405876\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	26.4612i	1.40838i	0.710009	+	0.704192i	$0.248691\pi$
$353$	26.4612i	1.40838i	−0.710009	+	0.704192i	$0.751309\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	24.7007	1.30365	0.651826	−	0.758369i	$-0.274003\pi$
$359$	24.7007	1.30365	0.651826	+	0.758369i	$0.274003\pi$
$360$	0	0
$361$	−18.8946	−0.994453
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	15.1349i	0.790035i	0.918674	+	0.395017i	$0.129261\pi$
$367$	15.1349i	0.790035i	−0.918674	+	0.395017i	$0.870739\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6.00852	−0.311947
$372$	0	0
$373$	− 8.94947i	− 0.463386i	−0.972789	−	0.231693i	$-0.925574\pi$
$373$	− 8.94947i	− 0.463386i	0.972789	−	0.231693i	$-0.0744265\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 44.4830i	− 2.29099i
$378$	0	0
$379$	−7.22270	−0.371005	−0.185503	−	0.982644i	$-0.559391\pi$
$379$	−7.22270	−0.371005	−0.185503	+	0.982644i	$0.559391\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 35.7267i	− 1.82555i	−0.408461	−	0.912776i	$-0.633934\pi$
$383$	− 35.7267i	− 1.82555i	0.408461	−	0.912776i	$-0.366066\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−0.706501	−0.0358210	−0.0179105	−	0.999840i	$-0.505701\pi$
$389$	−0.706501	−0.0358210	−0.0179105	+	0.999840i	$0.505701\pi$
$390$	0	0
$391$	−5.00018	−0.252870
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	6.44014i	0.323222i	0.986855	+	0.161611i	$0.0516689\pi$
$397$	6.44014i	0.323222i	−0.986855	+	0.161611i	$0.948331\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−26.7201	−1.33434	−0.667168	−	0.744907i	$-0.732494\pi$
$401$	−26.7201	−1.33434	−0.667168	+	0.744907i	$0.732494\pi$
$402$	0	0
$403$	21.3263i	1.06234i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 2.27830i	− 0.112931i
$408$	0	0
$409$	−24.0927	−1.19131	−0.595653	−	0.803242i	$-0.703107\pi$
$409$	−24.0927	−1.19131	−0.595653	+	0.803242i	$0.703107\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 5.99817i	− 0.295151i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−23.9133	−1.16824	−0.584120	−	0.811668i	$-0.698560\pi$
$419$	−23.9133	−1.16824	−0.584120	+	0.811668i	$0.698560\pi$
$420$	0	0
$421$	−9.50933	−0.463456	−0.231728	−	0.972781i	$-0.574438\pi$
$421$	−9.50933	−0.463456	−0.231728	+	0.972781i	$0.574438\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6.78417i	0.328309i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−29.8287	−1.43680	−0.718399	−	0.695632i	$-0.755125\pi$
$431$	−29.8287	−1.43680	−0.718399	+	0.695632i	$0.755125\pi$
$432$	0	0
$433$	− 14.2385i	− 0.684256i	−0.939653	−	0.342128i	$-0.888852\pi$
$433$	− 14.2385i	− 0.684256i	0.939653	−	0.342128i	$-0.111148\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.23015i	− 0.0588459i
$438$	0	0
$439$	32.9527	1.57275	0.786374	−	0.617751i	$-0.211956\pi$
$439$	32.9527	1.57275	0.786374	+	0.617751i	$0.211956\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	34.4224i	1.63546i	0.575604	+	0.817728i	$0.304767\pi$
$443$	34.4224i	1.63546i	−0.575604	+	0.817728i	$0.695233\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2.91326	0.137485	0.0687426	−	0.997634i	$-0.478101\pi$
$449$	2.91326	0.137485	0.0687426	+	0.997634i	$0.478101\pi$
$450$	0	0
$451$	12.0152	0.565774
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 23.8289i	− 1.11467i	−0.830288	−	0.557334i	$-0.811824\pi$
$457$	− 23.8289i	− 1.11467i	0.830288	−	0.557334i	$-0.188176\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.32644	0.294652	0.147326	−	0.989088i	$-0.452933\pi$
$461$	6.32644	0.294652	0.147326	+	0.989088i	$0.452933\pi$
$462$	0	0
$463$	4.31777i	0.200664i	0.994954	+	0.100332i	$0.0319905\pi$
$463$	4.31777i	0.200664i	−0.994954	+	0.100332i	$0.968010\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0.352336i	0.0163042i	0.999967	+	0.00815208i	$0.00259492\pi$
$467$	0.352336i	0.0163042i	−0.999967	+	0.00815208i	$0.997405\pi$
$468$	0	0
$469$	2.84157	0.131212
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 12.2504i	− 0.563274i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−28.0740	−1.28274	−0.641368	−	0.767234i	$-0.721633\pi$
$479$	−28.0740	−1.28274	−0.641368	+	0.767234i	$0.721633\pi$
$480$	0	0
$481$	6.97554	0.318057
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 27.9426i	− 1.26620i	−0.774070	−	0.633099i	$-0.781782\pi$
$487$	− 27.9426i	− 1.26620i	0.774070	−	0.633099i	$-0.218218\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−12.5303	−0.565486	−0.282743	−	0.959196i	$-0.591244\pi$
$491$	−12.5303	−0.565486	−0.282743	+	0.959196i	$0.591244\pi$
$492$	0	0
$493$	− 11.4088i	− 0.513827i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 0.606386i	− 0.0272001i
$498$	0	0
$499$	−42.8700	−1.91912	−0.959562	−	0.281498i	$-0.909169\pi$
$499$	−42.8700	−1.91912	−0.959562	+	0.281498i	$0.909169\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	12.8635i	0.573554i	0.957997	+	0.286777i	$0.0925838\pi$
$503$	12.8635i	0.573554i	−0.957997	+	0.286777i	$0.907416\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−13.6527	−0.605146	−0.302573	−	0.953126i	$-0.597846\pi$
$509$	−13.6527	−0.605146	−0.302573	+	0.953126i	$0.597846\pi$
$510$	0	0
$511$	−5.32119	−0.235396
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	21.7874i	0.958209i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	32.8962	1.44121	0.720605	−	0.693346i	$-0.243864\pi$
$521$	32.8962	1.44121	0.720605	+	0.693346i	$0.243864\pi$
$522$	0	0
$523$	− 18.0641i	− 0.789887i	−0.918705	−	0.394944i	$-0.870764\pi$
$523$	− 18.0641i	− 0.789887i	0.918705	−	0.394944i	$-0.129236\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.46967i	0.238262i
$528$	0	0
$529$	8.64165	0.375724
$530$	0	0
$531$	0	0
$532$	0	0
$533$	36.7874i	1.59344i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−10.9850	−0.473157
$540$	0	0
$541$	−2.25204	−0.0968226	−0.0484113	−	0.998827i	$-0.515416\pi$
$541$	−2.25204	−0.0968226	−0.0484113	+	0.998827i	$0.515416\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 8.79431i	− 0.376018i	−0.982167	−	0.188009i	$-0.939797\pi$
$547$	− 8.79431i	− 0.376018i	0.982167	−	0.188009i	$-0.0602033\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.80680	0.119574
$552$	0	0
$553$	− 6.57018i	− 0.279392i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 24.4199i	− 1.03470i	−0.855773	−	0.517352i	$-0.826918\pi$
$557$	− 24.4199i	− 1.03470i	0.855773	−	0.517352i	$-0.173082\pi$
$558$	0	0
$559$	37.5075	1.58640
$560$	0	0
$561$	0	0
$562$	0	0
$563$	10.1568i	0.428057i	0.976827	+	0.214029i	$0.0686586\pi$
$563$	10.1568i	0.428057i	−0.976827	+	0.214029i	$0.931341\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−45.7074	−1.91615	−0.958076	−	0.286514i	$-0.907504\pi$
$569$	−45.7074	−1.91615	−0.958076	+	0.286514i	$0.907504\pi$
$570$	0	0
$571$	10.7326	0.449144	0.224572	−	0.974457i	$-0.427902\pi$
$571$	10.7326	0.449144	0.224572	+	0.974457i	$0.427902\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	15.9857i	0.665493i	0.943016	+	0.332746i	$0.107975\pi$
$577$	15.9857i	0.665493i	−0.943016	+	0.332746i	$0.892025\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	3.30455	0.137096
$582$	0	0
$583$	− 14.8390i	− 0.614570i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	36.3465i	1.50018i	0.661335	+	0.750090i	$0.269990\pi$
$587$	36.3465i	1.50018i	−0.661335	+	0.750090i	$0.730010\pi$
$588$	0	0
$589$	−1.34565	−0.0554465
$590$	0	0
$591$	0	0
$592$	0	0
$593$	9.27142i	0.380732i	0.981713	+	0.190366i	$0.0609674\pi$
$593$	9.27142i	0.380732i	−0.981713	+	0.190366i	$0.939033\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−16.2328	−0.663256	−0.331628	−	0.943410i	$-0.607598\pi$
$599$	−16.2328	−0.663256	−0.331628	+	0.943410i	$0.607598\pi$
$600$	0	0
$601$	15.3885	0.627712	0.313856	−	0.949471i	$-0.398379\pi$
$601$	15.3885	0.627712	0.313856	+	0.949471i	$0.398379\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	9.64077i	0.391307i	0.980673	+	0.195653i	$0.0626827\pi$
$607$	9.64077i	0.391307i	−0.980673	+	0.195653i	$0.937317\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−66.7074	−2.69869
$612$	0	0
$613$	40.2967i	1.62757i	0.581166	+	0.813785i	$0.302597\pi$
$613$	40.2967i	1.62757i	−0.581166	+	0.813785i	$0.697403\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 15.2696i	− 0.614731i	−0.951592	−	0.307365i	$-0.900553\pi$
$617$	− 15.2696i	− 0.614731i	0.951592	−	0.307365i	$-0.0994474\pi$
$618$	0	0
$619$	26.0085	1.04537	0.522685	−	0.852526i	$-0.324930\pi$
$619$	26.0085	1.04537	0.522685	+	0.852526i	$0.324930\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 11.8914i	− 0.476418i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.78906	0.0713344
$630$	0	0
$631$	−4.84484	−0.192870	−0.0964350	−	0.995339i	$-0.530744\pi$
$631$	−4.84484	−0.192870	−0.0964350	+	0.995339i	$0.530744\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 33.6331i	− 1.33259i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	13.3263	0.526356	0.263178	−	0.964747i	$-0.415229\pi$
$641$	13.3263	0.526356	0.263178	+	0.964747i	$0.415229\pi$
$642$	0	0
$643$	36.0202i	1.42050i	0.703950	+	0.710250i	$0.251418\pi$
$643$	36.0202i	1.42050i	−0.703950	+	0.710250i	$0.748582\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 4.82219i	− 0.189580i	−0.995497	−	0.0947899i	$-0.969782\pi$
$647$	− 4.82219i	− 0.189580i	0.995497	−	0.0947899i	$-0.0302179\pi$
$648$	0	0
$649$	14.8135	0.581480
$650$	0	0
$651$	0	0
$652$	0	0
$653$	42.2723i	1.65424i	0.562024	+	0.827121i	$0.310023\pi$
$653$	42.2723i	1.65424i	−0.562024	+	0.827121i	$0.689977\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−37.5091	−1.46115	−0.730574	−	0.682834i	$-0.760747\pi$
$659$	−37.5091	−1.46115	−0.730574	+	0.682834i	$0.760747\pi$
$660$	0	0
$661$	−6.35905	−0.247338	−0.123669	−	0.992324i	$-0.539466\pi$
$661$	−6.35905	−0.247338	−0.123669	+	0.992324i	$0.539466\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 32.7612i	− 1.26852i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−16.7547	−0.646806
$672$	0	0
$673$	− 0.899676i	− 0.0346799i	−0.999850	−	0.0173400i	$-0.994480\pi$
$673$	− 0.899676i	− 0.0346799i	0.999850	−	0.0173400i	$-0.00551976\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 32.0288i	− 1.23097i	−0.788150	−	0.615483i	$-0.788961\pi$
$677$	− 32.0288i	− 1.23097i	0.788150	−	0.615483i	$-0.211039\pi$
$678$	0	0
$679$	−6.07911	−0.233295
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 14.7371i	− 0.563899i	−0.959429	−	0.281950i	$-0.909019\pi$
$683$	− 14.7371i	− 0.563899i	0.959429	−	0.281950i	$-0.0909811\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	45.4332	1.73087
$690$	0	0
$691$	−22.5437	−0.857603	−0.428802	−	0.903399i	$-0.641064\pi$
$691$	−22.5437	−0.857603	−0.428802	+	0.903399i	$0.641064\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	9.43508i	0.357379i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	19.9984	0.755327	0.377664	−	0.925943i	$-0.376728\pi$
$701$	19.9984	0.755327	0.377664	+	0.925943i	$0.376728\pi$
$702$	0	0
$703$	0.440144i	0.0166004i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 7.38711i	− 0.277821i
$708$	0	0
$709$	−15.0170	−0.563976	−0.281988	−	0.959418i	$-0.590994\pi$
$709$	−15.0170	−0.563976	−0.281988	+	0.959418i	$0.590994\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	15.7065i	0.588213i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	25.8289	0.963254	0.481627	−	0.876376i	$-0.340046\pi$
$719$	25.8289	0.963254	0.481627	+	0.876376i	$0.340046\pi$
$720$	0	0
$721$	7.94877	0.296028
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	10.6266i	0.394120i	0.980391	+	0.197060i	$0.0631394\pi$
$727$	10.6266i	0.394120i	−0.980391	+	0.197060i	$0.936861\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9.61976	0.355800
$732$	0	0
$733$	41.7613i	1.54249i	0.636538	+	0.771245i	$0.280366\pi$
$733$	41.7613i	1.54249i	−0.636538	+	0.771245i	$0.719634\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.01774i	0.258502i
$738$	0	0
$739$	−49.1030	−1.80628	−0.903141	−	0.429343i	$-0.858745\pi$
$739$	−49.1030	−1.80628	−0.903141	+	0.429343i	$0.858745\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 19.3111i	− 0.708454i	−0.935159	−	0.354227i	$-0.884744\pi$
$743$	− 19.3111i	− 0.708454i	0.935159	−	0.354227i	$-0.115256\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3.48072	−0.127183
$750$	0	0
$751$	8.31451	0.303401	0.151700	−	0.988427i	$-0.451525\pi$
$751$	8.31451	0.303401	0.151700	+	0.988427i	$0.451525\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	33.8667i	1.23091i	0.788173	+	0.615453i	$0.211027\pi$
$757$	33.8667i	1.23091i	−0.788173	+	0.615453i	$0.788973\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	19.0481	0.690495	0.345247	−	0.938512i	$-0.387795\pi$
$761$	19.0481	0.690495	0.345247	+	0.938512i	$0.387795\pi$
$762$	0	0
$763$	− 4.87940i	− 0.176646i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	45.3550i	1.63767i
$768$	0	0
$769$	−33.6627	−1.21391	−0.606953	−	0.794738i	$-0.707608\pi$
$769$	−33.6627	−1.21391	−0.606953	+	0.794738i	$0.707608\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 23.5354i	− 0.846509i	−0.906011	−	0.423254i	$-0.860888\pi$
$773$	− 23.5354i	− 0.846509i	0.906011	−	0.423254i	$-0.139112\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.32122	−0.0831663
$780$	0	0
$781$	1.49757	0.0535873
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 23.5321i	− 0.838830i	−0.907795	−	0.419415i	$-0.862235\pi$
$787$	− 23.5321i	− 0.838830i	0.907795	−	0.419415i	$-0.137765\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−12.0067	−0.426909
$792$	0	0
$793$	− 51.2984i	− 1.82166i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 2.53052i	− 0.0896355i	−0.998995	−	0.0448177i	$-0.985729\pi$
$797$	− 2.53052i	− 0.0896355i	0.998995	−	0.0448177i	$-0.0142707\pi$
$798$	0	0
$799$	−17.1088	−0.605266
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 13.1416i	− 0.463756i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	21.5178	0.756526	0.378263	−	0.925698i	$-0.376521\pi$
$809$	21.5178	0.756526	0.378263	+	0.925698i	$0.376521\pi$
$810$	0	0
$811$	−45.0566	−1.58215	−0.791076	−	0.611717i	$-0.790479\pi$
$811$	−45.0566	−1.58215	−0.791076	+	0.611717i	$0.790479\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.36666i	0.0827988i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−29.8095	−1.04036	−0.520179	−	0.854057i	$-0.674135\pi$
$821$	−29.8095	−1.04036	−0.520179	+	0.854057i	$0.674135\pi$
$822$	0	0
$823$	− 12.9122i	− 0.450091i	−0.974348	−	0.225045i	$-0.927747\pi$
$823$	− 12.9122i	− 0.450091i	0.974348	−	0.225045i	$-0.0722530\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 8.92395i	− 0.310316i	−0.987890	−	0.155158i	$-0.950411\pi$
$827$	− 8.92395i	− 0.310316i	0.987890	−	0.155158i	$-0.0495887\pi$
$828$	0	0
$829$	−23.2259	−0.806670	−0.403335	−	0.915052i	$-0.632149\pi$
$829$	−23.2259	−0.806670	−0.403335	+	0.915052i	$0.632149\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 8.62608i	− 0.298876i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	26.3677	0.910315	0.455157	−	0.890411i	$-0.349583\pi$
$839$	26.3677	0.910315	0.455157	+	0.890411i	$0.349583\pi$
$840$	0	0
$841$	45.7505	1.57760
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 5.56328i	− 0.191157i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	5.13739	0.176108
$852$	0	0
$853$	− 54.9223i	− 1.88050i	−0.340480	−	0.940252i	$-0.610589\pi$
$853$	− 54.9223i	− 1.88050i	0.340480	−	0.940252i	$-0.389411\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 7.93933i	− 0.271202i	−0.990763	−	0.135601i	$-0.956703\pi$
$857$	− 7.93933i	− 0.271202i	0.990763	−	0.135601i	$-0.0432966\pi$
$858$	0	0
$859$	18.8769	0.644070	0.322035	−	0.946728i	$-0.395633\pi$
$859$	18.8769	0.644070	0.322035	+	0.946728i	$0.395633\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 19.4215i	− 0.661116i	−0.943786	−	0.330558i	$-0.892763\pi$
$863$	− 19.4215i	− 0.661116i	0.943786	−	0.330558i	$-0.107237\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	16.2261	0.550434
$870$	0	0
$871$	−21.4865	−0.728042
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 12.3811i	− 0.418080i	−0.977907	−	0.209040i	$-0.932966\pi$
$877$	− 12.3811i	− 0.418080i	0.977907	−	0.209040i	$-0.0670339\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	16.8910	0.569072	0.284536	−	0.958665i	$-0.408160\pi$
$881$	16.8910	0.569072	0.284536	+	0.958665i	$0.408160\pi$
$882$	0	0
$883$	− 17.8457i	− 0.600556i	−0.953852	−	0.300278i	$-0.902921\pi$
$883$	− 17.8457i	− 0.600556i	0.953852	−	0.300278i	$-0.0970794\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 6.98730i	− 0.234611i	−0.993096	−	0.117305i	$-0.962574\pi$
$887$	− 6.98730i	− 0.234611i	0.993096	−	0.117305i	$-0.0374256\pi$
$888$	0	0
$889$	−2.13306	−0.0715406
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 4.20911i	− 0.140853i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−35.8372	−1.19524
$900$	0	0
$901$	11.6525	0.388202
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	36.0085i	1.19564i	0.801630	+	0.597821i	$0.203967\pi$
$907$	36.0085i	1.19564i	−0.801630	+	0.597821i	$0.796033\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−9.41987	−0.312094	−0.156047	−	0.987750i	$-0.549875\pi$
$911$	−9.41987	−0.312094	−0.156047	+	0.987750i	$0.549875\pi$
$912$	0	0
$913$	8.16115i	0.270095i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.52200i	0.182353i
$918$	0	0
$919$	9.54389	0.314824	0.157412	−	0.987533i	$-0.449685\pi$
$919$	9.54389	0.314824	0.157412	+	0.987533i	$0.449685\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4.58517i	0.150923i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	8.01939	0.263108	0.131554	−	0.991309i	$-0.458003\pi$
$929$	8.01939	0.263108	0.131554	+	0.991309i	$0.458003\pi$
$930$	0	0
$931$	2.12219	0.0695520
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 36.1070i	− 1.17956i	−0.807563	−	0.589782i	$-0.799214\pi$
$937$	− 36.1070i	− 1.17956i	0.807563	−	0.589782i	$-0.200786\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	20.8153	0.678560	0.339280	−	0.940685i	$-0.389817\pi$
$941$	20.8153	0.678560	0.339280	+	0.940685i	$0.389817\pi$
$942$	0	0
$943$	27.0934i	0.882283i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.92413i	− 0.160013i	−0.996794	−	0.0800064i	$-0.974506\pi$
$947$	− 4.92413i	− 0.160013i	0.996794	−	0.0800064i	$-0.0254941\pi$
$948$	0	0
$949$	40.2361	1.30612
$950$	0	0
$951$	0	0
$952$	0	0
$953$	6.23702i	0.202037i	0.994885	+	0.101018i	$0.0322101\pi$
$953$	6.23702i	0.202037i	−0.994885	+	0.101018i	$0.967790\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4.68894	0.151414
$960$	0	0
$961$	−13.8188	−0.445767
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	17.7648i	0.571277i	0.958337	+	0.285639i	$0.0922057\pi$
$967$	17.7648i	0.571277i	−0.958337	+	0.285639i	$0.907794\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	9.98061	0.320293	0.160147	−	0.987093i	$-0.448803\pi$
$971$	9.98061	0.320293	0.160147	+	0.987093i	$0.448803\pi$
$972$	0	0
$973$	13.7823i	0.441842i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 16.5328i	− 0.528932i	−0.964395	−	0.264466i	$-0.914804\pi$
$977$	− 16.5328i	− 0.528932i	0.964395	−	0.264466i	$-0.0851957\pi$
$978$	0	0
$979$	29.3677	0.938597
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 57.4418i	− 1.83211i	−0.401055	−	0.916054i	$-0.631356\pi$
$983$	− 57.4418i	− 1.83211i	0.401055	−	0.916054i	$-0.368644\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	27.6238	0.878384
$990$	0	0
$991$	55.5294	1.76395	0.881975	−	0.471297i	$-0.156214\pi$
$991$	55.5294	1.76395	0.881975	+	0.471297i	$0.156214\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 8.11969i	− 0.257153i	−0.991700	−	0.128577i	$-0.958959\pi$
$997$	− 8.11969i	− 0.257153i	0.991700	−	0.128577i	$-0.0410408\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8100.2.d.q.649.6		8
3.2	odd	2		8100.2.d.s.649.6		8
5.2	odd	4		8100.2.a.z.1.2		4
5.3	odd	4		8100.2.a.x.1.3		4
5.4	even	2	inner	8100.2.d.q.649.3		8
9.2	odd	6		2700.2.s.d.1549.6		16
9.4	even	3		900.2.s.d.349.1		16
9.5	odd	6		2700.2.s.d.2449.3		16
9.7	even	3		900.2.s.d.49.8		16
15.2	even	4		8100.2.a.ba.1.2		4
15.8	even	4		8100.2.a.y.1.3		4
15.14	odd	2		8100.2.d.s.649.3		8
45.2	even	12		2700.2.i.d.901.3		8
45.4	even	6		900.2.s.d.349.8		16
45.7	odd	12		900.2.i.e.301.2	yes	8
45.13	odd	12		900.2.i.d.601.3	yes	8
45.14	odd	6		2700.2.s.d.2449.6		16
45.22	odd	12		900.2.i.e.601.2	yes	8
45.23	even	12		2700.2.i.e.1801.2		8
45.29	odd	6		2700.2.s.d.1549.3		16
45.32	even	12		2700.2.i.d.1801.3		8
45.34	even	6		900.2.s.d.49.1		16
45.38	even	12		2700.2.i.e.901.2		8
45.43	odd	12		900.2.i.d.301.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.2.i.d.301.3	✓	8	45.43	odd	12
900.2.i.d.601.3	yes	8	45.13	odd	12
900.2.i.e.301.2	yes	8	45.7	odd	12
900.2.i.e.601.2	yes	8	45.22	odd	12
900.2.s.d.49.1		16	45.34	even	6
900.2.s.d.49.8		16	9.7	even	3
900.2.s.d.349.1		16	9.4	even	3
900.2.s.d.349.8		16	45.4	even	6
2700.2.i.d.901.3		8	45.2	even	12
2700.2.i.d.1801.3		8	45.32	even	12
2700.2.i.e.901.2		8	45.38	even	12
2700.2.i.e.1801.2		8	45.23	even	12
2700.2.s.d.1549.3		16	45.29	odd	6
2700.2.s.d.1549.6		16	9.2	odd	6
2700.2.s.d.2449.3		16	9.5	odd	6
2700.2.s.d.2449.6		16	45.14	odd	6
8100.2.a.x.1.3		4	5.3	odd	4
8100.2.a.y.1.3		4	15.8	even	4
8100.2.a.z.1.2		4	5.2	odd	4
8100.2.a.ba.1.2		4	15.2	even	4
8100.2.d.q.649.3		8	5.4	even	2	inner
8100.2.d.q.649.6		8	1.1	even	1	trivial
8100.2.d.s.649.3		8	15.14	odd	2
8100.2.d.s.649.6		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8100.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.680426i q^{7} -1.68043 q^{11} -5.14503i q^{13} -1.31957i q^{17} +0.324642 q^{19} -3.78924i q^{23} +8.64584 q^{29} -4.14503 q^{31} +1.35578i q^{37} -7.15009 q^{41} +7.29005i q^{43} -12.9654i q^{47} +6.53702 q^{49} +8.83052i q^{53} -8.81532 q^{59} +9.97048 q^{61} -4.17617i q^{67} -0.891185 q^{71} +7.82038i q^{73} -1.14341i q^{77} -9.65597 q^{79} -4.85659i q^{83} -17.4764 q^{89} +3.50081 q^{91} +8.93427i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{11} + 8 q^{19} - 18 q^{29} + 4 q^{31} - 18 q^{41} - 18 q^{49} - 30 q^{59} - 2 q^{61} - 24 q^{71} + 14 q^{79} - 6 q^{89} - 22 q^{91}+O(q^{100}) \) 8 * q - 6 * q^11 + 8 * q^19 - 18 * q^29 + 4 * q^31 - 18 * q^41 - 18 * q^49 - 30 * q^59 - 2 * q^61 - 24 * q^71 + 14 * q^79 - 6 * q^89 - 22 * q^91

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